AP Calculus AB – Continuity and Intermediate Value Theorem
When is a function
continuous?
A function is continuous over its domain if you can draw the graph of the
function without „lifting your pencil.‟
Definition of
Continuity at a
Point
A function, f, is continuous at a point, c, if:
1. f(c) is defined.
2.
exists.
3. #1 = #2
Do you need all three conditions:
Examples – page 71
Definition of
Continuity on a
Closed Interval
Describe where
this function is
continuous
Intermediate Value
Theorem (IVT)
- An “existence”
theorem
A function, f, is continuous on the closed interval [a,b] if it is
continuous on (a,b) and f is continuous from the right at a and f
is continuous from the left at b.
f(x) = tan x
If f is continuous on a closed interval [a,b] and f(a) f(b) and k
is any number BETWEEN f(a) and f(b), then there EXISTS at
least one number, c, in [a,b] such that f(c) = k.
Example:
f(x) = x2 – 6x + 8
Does the Intermediate Value Theorem apply to the interval [0,3] to get a
value of f(c) = 1? Find the value of c.